I don't think that particular implementation of a "Folder Browse" dialog can include a textbox. If I'm interpreting it correctly, it's part of the OS and a textbox isn't one of it's features if memory serves.

However, as far as the initial path being ????? or always remembering the last path used, I think is possible but it remembering every last path based on context from where it is being opened feature wise, potentially more work. Last path seems most intelligent... Since Reaper deals with projects and media, it can't do the obviously simple fix of always opening where reaper.exe is (which would be as useless) so last path used it is. IOW, Reaper would need to be aware of where you 'want' it to open by default, your project folders, sample folders, who knows.

It's also important to remember that since Reaper tries to stay compatible both cross-platform (three of them mind you) and more importantly on much older OS versions all the way back to XP, it requires avoiding more modern OS implementations/features.

So, part of what you want probably is entirely possible, being and easy 10 minutes worth of code, I highly, highly doubt as changing it likely requires a number of other things to be considered and/or addressed that on the surface aren't obvious. What you see it doing and what it took to make it do that are typically not a 1:1 match complexity wise. Based on one of the links in your signature, I assume you already know most of this.